Transcription of Variance, covariance, correlation, moment-generating functions
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Variance, covariance, correlation, moment-generating functions
Math 461 Introduction to HildebrandVariance, covariance, correlation, moment-generating functions [In the Ross text, this is covered in Sections and See also the Chapter Summary on pp. 405 407.] Variance: Definition:Var(X) = E(X2) E(X)2(=E(X E(X))2) Properties:Var(c) = 0, Var(cX) =c2 Var(X), Var(X+c) = Var(X) Covariance: Definition:Cov(X,Y) =E(XY) E(X)E(Y)(=E(X E(X))(Y E(Y))) Properties: Symmetry:Cov(X,Y) = Cov(Y,X) Relation to variance:Var(X) = Cov(X,X), Var(X+Y) = Var(X)+Var(Y)+2 Cov(X,Y) Bilinearity:Cov(cX,Y) = Cov(X,cY) =cCov(X,Y),Cov(X1+X2,Y) = Cov(X1,Y) + Cov(X2,Y),Cov(X,Y1+Y2) = Cov(X,Y1) + Cov(X,Y2). Product formula:Cov( ni=1Xi, mj=1Yj) = ni=1 my=1 Cov(Xi,Yj) Correlation: Definition: (X,Y) =Cov(X,Y) Var(X) Var(Y) Properties: 1 (X,Y) 1 moment-generating function: Definition:M(t) =MX(t) = E(etX) Computing moments via mgf s:The derivates ofM(t), evaluated att= 0, give the successive moments of a random variableX:M(0) = 1,M (0) = E(X),M (0) = E(X2),M (0) = E(X3),etc.
• While for independent r.v.’s, covariance and correlation are always 0, the converse is not true: One can construct r.v.’s X and Y that have 0 covariance/correlation 0 (“uncorrelated”), but which are not independent. 2
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